Radiographic imaging such as x-ray imaging has been used for years in medical applications and for non-destructive testing.
Normally, an x-ray imaging system includes an x-ray source and an x-ray detector consisting of multiple detector elements. The x-ray source emits x-rays, which pass through a subject or object to be imaged and are then registered by the detector. Since some materials absorb a larger fraction of the x-rays than others, an image is formed of the interior of the subject or object.
An x-ray computed tomography (CT) system includes an x-ray source and an x-ray detector arranged in such a way that projection images of the subject or object can be acquired in different view angles covering at least 180 degrees. This is most commonly achieved by mounting the source and detector on a support that is able to rotate around the subject or object. An image containing the projections registered in the different detector elements for the different view angles is called a sinogram. In the following, a collection of projections registered in the different detector elements for different view angles will be referred to as a sinogram even if the detector is two-dimensional, making the sinogram a three-dimensional image.
A further development of x-ray imaging is energy-resolved x-ray imaging, also known as spectral x-ray imaging, where the x-ray transmission is measured for several different energy levels. This can be achieved by letting the source switch rapidly between two different emission spectra, by using two or more x-ray sources emitting different x-ray spectra, or by using an energy-discriminating detector which measures the incoming radiation in two or more energy levels. One example of such a detector is a multi-bin photon-counting detector, where each registered photon generates a current pulse which is compared to a set of thresholds, thereby counting the number of photons incident in each of a number of energy bins.
A spectral x-ray projection measurement results in one projection image for each energy level. A weighted sum of these can be made to optimize the contrast-to-noise ratio (CNR) for a specified imaging task as described in Tapiovaara and Wagner, “SNR and DQE analysis of broad spectrum X-ray imaging”, Phys. Med. Biol. 30, 519.
Another technique enabled by energy-resolved x-ray imaging is basis material decomposition. This technique utilizes the fact that all substances built up from elements with low atomic number, such as human tissue, have linear attenuation coefficients μ(E) whose energy dependence can be expressed, to a good approximation, as a linear combination of two basis functions:μ(E)=a1f1(E)+a2f2(E).where fi are the basis functions and ai are the corresponding basis coefficients. If there is one or more element in the imaged volume with high atomic number, high enough for a k-absorption edge to be present in the energy range used for the imaging, one basis function must be added for each such element. In the field of medical imaging, such k-edge elements can typically be iodine or gadolinium, substances that are used as contrast agents.
Basis material decomposition has been described in Alvarez and Macovski, “Energy-selective reconstructions in X-ray computerised tomography”, Phys. Med. Biol. 21, 733. In basis material decomposition, the integral of each of the basis coefficients, Ai=∫l aidl for i=1, . . . , N where N is the number of basis functions, is inferred from the measured data in each projection ray l from the source to a detector element. In one implementation, this is accomplished by first expressing the expected registered number of counts in each energy bin as a function of Ai:
      λ    i    =            ∫              E        =        0            ∞        ⁢                            S          i                ⁡                  (          E          )                    ⁢              exp        ⁡                  (                      -                                          ∑                                  j                  =                  1                                N                            ⁢                                                          ⁢                                                A                  j                                ⁢                                                      f                    j                                    ⁡                                      (                    E                    )                                                                                )                    ⁢      dE      
Here, λi is the expected number of counts in energy bin i, E is the energy, Si is a response function which depends on the spectrum shape incident on the imaged object, the quantum efficiency of the detector and the sensitivity of energy bin i to x-rays with energy E. Even though the term “energy bin” is most commonly used for photon-counting detectors, this formula can also describe other energy resolving x-ray systems such as multi-layer detectors or kVp switching sources.
Then, the maximum likelihood method may be used to estimate Ai, under the assumption that the number of counts in each bin is a Poisson distributed random variable. This is accomplished by minimizing the negative log-likelihood function, see Roessl and Proksa, K-edge imaging in x-ray computed tomography using multi-bin photon counting detectors, Phys. Med. Biol. 52 (2007), 4679-4696:
            A      ^        1    ,  …  ⁢          ,                    A        ^            N        =                                        arg            ⁢                                                  ⁢            min                                              A              1                        ,                                                  ⁢            …            ⁢                                                  ,                                                  ⁢                          A              N                                      ⁢                              ∑                          i              =              1                                      M              b                                ⁢                                          ⁢                                    λ              i                        ⁡                          (                                                A                  1                                ,                                                                  ⁢                …                ⁢                                                                  ,                                                                  ⁢                                  A                  N                                            )                                          -                        m          i                ⁢                                  ⁢        ln        ⁢                                  ⁢                              λ            i                    ⁡                      (                                          A                1                            ,                                                          ⁢              …              ⁢                                                          ,                                                          ⁢                              A                N                                      )                              where mi is the number of measured counts in energy bin i and Mb is the number of energy bins.
When the resulting estimated basis coefficient line integral Âi for each projection line is arranged into an image matrix, the result is a material specific projection image, also called a basis image, for each basis i. This basis image can either be viewed directly (e.g. in projection x-ray imaging) or taken as input to a reconstruction algorithm to form maps of basis coefficients ai inside the object (e.g. in CT). Anyway, the result of a basis decomposition can be regarded as one or more basis image representations, such as the basis coefficient line integrals or the basis coefficients themselves.
However, a well-known limitation of this technique is that the variance of the estimated line integrals normally increases with the number of bases used in the basis decomposition. Among other things, this results in an unfortunate trade-off between improved tissue quantification and increased image noise.